Biology Reference
In-Depth Information
CHAPTER
7
Partial
Least Squares A
nalysis
Partial Least Squares (PLS) is a method for exploring patterns of covariation between
two (and potentially more) blocks of variables. It can be used to study covariation between
shape and environmental variables, such as between nasal cavity morphology and temper-
ature and aridity (
Noback et al., 2011
) or between shape and a collection of climatic, geo-
graphic and biotic variables, e.g. vegetation type and human density (
Monteiro et al.,
2003
). PLS can also be used to analyze the relationship between form and function, such
as the relationship between horn morphology and fighting behavior in bovids (
Lundrigan,
1996
) or between morphology and disease status (
Lowe et al., 1997; Bookstein et al., 2002
).
As well as being useful for analyzing the relationship between shape and non-shape vari-
ables, PLS can also be used to analyze the covariances between two or more blocks of
shape variables. That ability to examine relationships between two or more blocks of shape
variables makes PLS useful for synthesizing information about three-dimensional
morphologies from two two-dimensional views (e.g.
Rohlf and Corti, 2000
) or for relating
the shapes of functionally interacting parts such as the maxillary and mandibular denti-
tions (Sheets et al., in press). Because PLS can be used to examine the covariance between
blocks of shape variables, the method can be used to examine morphological integration
and modularity (
Bookstein et al., 2003; Klingenberg et al., 2003; Bastir and Rosas, 2004;
Bastir et al., 2005; Mitteroecker and Bookstein, 2007
).
PLS, like the ordination methods discussed in the previous chapter, reduces the
dimensionality of the data (of both blocks), yielding axes that explain the covariance
between blocks, ordered from the pair that explains the maximal covariance to the pair
that explains the least, all of which are mutually orthogonal. It also gives scores on those
axes along with the proportion of the total covariance between blocks explained by that
pair of axes and the correlation between the scores for each pair of axes. It obviously dif-
fers from Principal Components Analysis (PCA) in that it examines the covariance
between blocks rather than the variance within a block. It also differs from other methods
that examine the relationship between sets of variables, for reasons that will be discussed
in more detail below. An important feature of PLS is that the variables within the blocks
need not be independent of each other. For example, in the study relating nasal cavity
